MATH 160 Study Guide for Chapter5 IntegrationSpring, 2008

This Study Guide describes everything from Chapter 5 that you will be expected to know, understand, and be able to do on Exam 3 and/or the final exam. Exam questions from Chapter 5 will ask you to do one or more of the tasks described in this Study Guide. Of course, to do the things listed here you must be able to do almost everything listed in previous Study Guides. Don’t be surprised if some exam questions require you to use ideas or skills from earlier in the course.

Sec 5.1 Estimating with Finite Sums

__1.Use finite sums to calculate reasonable estimates for

•the area of a region enclosed by the graph of a positive-valued function y =f(x), the x-axis, and vertical lines y = a and y = b, and

•the distance a body travels during a given time interval from a graph of its velocity.

Representative homeworkproblems: Select from pages 333 – 335; #1 – 8, 12. Page 388; #1, 2.

__2.• From a table giving the velocity of a moving body at various times (not necessarily equal spaced), calculate a reasonable estimate for the displacement (change in position) of the body over a given time interval.
(You are expected to use a calculator.)

• Given an equation v = f(t) for a velocity function (whose antiderivative can’t be found easily), generate a table that gives the velocity of the moving body at various (equally spaced) times, and use this table to calculate a reasonable estimate for the displacement of the body over a given time interval.
(You are expected to use a calculator.)

Explain why one might expect your calculations to produce a reasonable estimate for the displacement.

Representative homeworkproblems: Select from pages 333 –334; #9 – 14.

Sec 5.2 Sigma Notation and Limits of Finite Sums

__1.A sum of the form SP = is called a Riemann sum for f on the interval [a, b].
Use words and pictures to explain how the interval [a, b] enters into the Riemann sum and what each of the symbols n, , P, f, xk, xk, and ck in this expression means.

__2.Given a function y = f(x) defined on an interval [a, b] and given a partition a = x0x1 < … < xn-1xn = b of the interval, choose appropriate evaluation points c1, c2, c3, … , cn , write an expression for the associated Riemann sum (in expanded form), and calculate the numerical value of the Riemann sum.
(Proficiency with summation notation is not required, but expect to see summation notation in later courses.)

Representative homeworkproblems: Select from pages 342 – 343; #29 – 32.
Study suggestion: Calculator Lab on Riemann Sums.

__3.Write an expression/equation that defines the Riemann integral in terms of Riemann sums.
Explain (using words and pictures) what this definition means and why it is necessary to take the limit as n approaches infinity in this definition.
Explain why the variable of integration in is called a dummy variable.

__4.Use a Riemann sum to calculate a reasonably accurate approximate value for a given (definite) integral. (You are expected to use a calculator.)

Study suggestion: Calculator Lab on Riemann Sums.

Sec 5.3 The Definite Integral

__0.Use standard area formulas for triangles, rectangles, quarter circles, half circles, etc. to evaluate definite integrals by identifying the integral with the area of a familiar shape.

__1.Use properties of the definite integral summarized in Table 5.3 entries #3 - #5 to evaluate definite integrals by rewriting the integral as a combination of integrals that either have given values or can be evaluated by interpreting the integral as (positive or negative) area.

Representative homeworkproblems: Pg 353; select from odd numbered problems from #9 – 26.

__2.Estimate integrals by using inequalities #6 and #7 in Table 3.5.

Suggested problems: Select from page 355; #63 – 72. (In #63 & 64 assume a < b. In #68 the integral should be from 0 to 1, not 1 to 0.)

Sec 5.4 The Fundamental Theorem of Calculus

__1.Use a calculator to evaluate a given function F(x) = for various values of x.
Use the resulting values to sketch a graph of the function y = F(x).

Study suggestion: Calculator Lab on the Fundamental Theorem of Calculus.

__2.Use a calculator to compute an approximate value for the derivative F′(c) of a given function
F(x) = at a specified point x = c from the definition of the derivative of a function at a point.
Interpret the calculation in terms of the graph of the integrand y = f(x).

Study suggestion: Calculator Lab on the Fundamental Theorem of Calculus.

__3.Write a differentiation formula for functions F(x) = defined by an integral with continuous integrand. Explain why it is appropriate to call this theorem The Fundamental Theorem of Calculus. (This theorem is on page 358 of the textbook where it is called The Fundamental Theorem of Calculus Part 1). Use this formula to compute derivatives of functions defined by integrals with the independent variable as the upper limit of integration.

Representative homeworkproblems: Page 365; #31 & 32. Page 390; #77.

__4.Use the differentiation formula from #3 with the Chain Rule and properties of the integral to calculate the derivatives of functions F(x) = and F(x) = where the upper or lower limit of integration is a differentiable function y = g(x).

Representative homeworkproblems: Page 365; #27 – 36. Page 367; #63 & 64. Page 390; #78, 79, 80

__5.Use the differentiation formula from #3 (in reverse) to write an expression for an antiderivative of a given continuous function and to solve initial value problems (and to show your answers are correct).

Suggested problems: Select from page 366; # 47 – 54

__6.Explain the difference between an indefinite integral and a definite integral .
State a Theorem that gives a connection between indefinite integrals (antiderivatives) and definite integrals.
Show how this result follows from The Fundamental Theorem of Calculus Part 1 (from #3 above).
(In the textbook this result is called The Fundamental Theorem of Calculus Part 2.)

__7.Infer information about the motion of an object moving a long a coordinate axis from a graph of its velocity.

Representative homeworkproblems: Page 366; #59 & 60. Page 388; #1, 2.

__8.Use The Fundamental Theorem of Calculus Part 2 to evaluate definite integrals.

Suggested problems: Page 365; #1 – 26.

Sec 5.5 Indefinite Integrals and Substitution

__1.Write an integration formula corresponding to the Chain Rule.
Explain how integration by substitution is used to apply (or implement) this integration formula.

Study suggestion: Pages 368 – 371 of the text.
Representative homeworkproblems: Pages 374 – 375; #1 – 12. Page 376; #61, 62.

__2.Evaluate indefinite integrals by u-substitution.
After you have evaluated an indefinite integral by u-substitution, identify the functions f and g that connect your work with the Chain Rule. Express the u you used in the substitution process in terms of one or more of the functions f and g and their derivatives.

Representative homeworkproblems: Select from pages 374 – 375;#13 – 52.

__3.Solve initial value problems that require u-substitution method to find the required antiderivatives.

Representative homeworkproblems: Pages 375 – 376; #53 – 60.

NOTE:You will not be asked to derive or to remember the integration formulas for sin2θ and cos2θ (pg 373)on the Final Exam. These two integration formulas will be given if needed to answer an exam question.

Sec 5.6 Substitution and Area between Curves

__1.Use change of variables (u-substitution), including changes in the limits of integration, to evaluate definite integrals.

Representative homeworkproblems: Pages 383; #1 – 24.

__1.Rewrite definite integrals involving arbitrary functions (indicated only by function notation), including changes in the limits of integration, by making a change of variables (that is, a u-substitution). (You might or might not be able to evaluate the resulting integral.)

Representative homeworkproblems: Pages 386 – 387; #83, 84, 87, 88, 89.

__3.Use a definite integrals to calculate the area of a region enclosed by the graph of a given function y =f(x), the x-axis, and vertical lines y = a and y = b. (Compare item #1 under Sec. 5.3 above.)

Representative homeworkproblems: Page 365 – 366; #37 – 46 and #55. Page 384 – 385; #25 – 29.

__4.Given a description of a region bounded by the graphs of continuous functions and (perhaps) vertical and/or horizontal lines, find the area of the region by
• sketching the region and deciding whether it is better to integrate with respect to x or with respect to y,
• sketching a representative area element used to find the integral for the area,
• writing the definite integral (or integrals) that represents the area, and
• evaluating the integral(s).

Representative homeworkproblems: Pages 384 – 386; #30 – 80.

October, 2007